On Banach function M-spaces

Let ω be a measure space and L_∞ be the Banach algebra of all essentially bounded measurable scalar functions defined on ω. In the present paper we study so-called Banach M-spaces X such as Banach L_∞-modules of ^m-valued measurable functions in the vector-valued case m > 1. We present with complete proofs the new technical tool via so-called representative families (of m measurable functions) for X which is necessary for constructing correctly the theory of Banach M-spaces X of ^m-valued functions with m > 1. We emphasize main new moments in proving basic theorems for Banach M-spaces of ^m-valued functions with m > 1.